X-ray diffraction on thin films and multilayers

Transcription

1 Chapter 3 X-ray diffraction on thin films and multilayers 3.1 Introduction In this section we present ex-situ X-ray diffraction measurements as well as their numerical simulations for epitaxial CoO and MnO thin films and multilayers. In the first half of this chapter we show how information about thickness, strain and structure is obtained using conventional methods. The second half is devoted to the discovery of rather complicated diffuse scattering patterns near reciprocal lattice points, which are characteristic of the presence of misfit dislocations appearing in partially relaxed films of strained CoO and MnO films. Up till now such patterns have been reported in literature only in the case of semiconductors. Numerical simulations of the recorded diffracted intensity are presented, comparing a simple block model with more realistic models comprising the full displacement field of the misfit dislocations responsible for relaxation. It will become clear that X-ray diffraction is not only a sensitive tool to detect order, but, if carefully used, can give insight into the nature and magnitude of disorder related to the presence of misfit dislocations in strained epitaxial layers. 49

2 50 Chapter 3. X-ray diffraction on thin films and multilayers 3.2 X-ray diffraction versus RHEED In the previous chapter we were mainly concerned with the characterization of surfaces of crystalline substrates and thin films by the use of electron diffraction. We have seen that for these techniques a good vacuum environment is necessary which is an advantage in the case of thin films that are reactive in air (see for example the oxidation of MnO in oxygen atmosphere 2.12). In this chapter we will mainly focus on ex-situ characterization of surfaces by X-rays, therefore this is a good opportunity to shortly compare the two complementary techniques. With RHEED, diffracted electrons are easily detectable by the use of a fluorescent screen, and the small wavelength allows the visualization of a large area of the reciprocal lattice but restricts at the same time the precise determination of small differences in lattice constants. X-rays on the other hand can easily be monochromatized and the larger wavelength ( 1.54 Å) allows the precise determination of the lattice constants as well as the strain state of the thin films. The coherence length, the penetration depth as well as the dynamic range for detection are higher for X-rays than for RHEED. Hence X-ray diffraction is a valuable tool for the study of thin films and multilayers as long as the films are inert or can be capped with a protective layer which prevents the deterioration of the films in contact with air. While for electrons, which interact strongly with matter, only complicated dynamical theory calculations can be used to attain insight into material properties, for hard X-rays a simple kinematical approach is in most of the cases sufficient to fit the measured diffraction spectrum because their interaction with matter is weak. Table 3.1 presents an overview of the two techniques. Property Electrons in RHEED X-rays in diffractometer Interaction Strong Weak Dynamic range 10 3 (CCD) 10 6 Focusing Yes Yes Wavelangth (λ (in Å)) (Cu) Environment vacuum vacuum or air Theory Dynamic Kinematic Table 3.1: Overview of similarities and differences between the RHEED and X-ray diffraction techniques.

3 3.3. X-ray diffraction by crystals X-ray diffraction by crystals The interaction of hard X-rays with matter happens through the scattering by the electrons of the atomic constituents. Assuming the dipole approximation, the scattered wave with amplitude A 1 from an electron at position r e observed at a distance R 0, is given by the formula: A 1 = A 0 e 2 mc 2 1 R 0 e [i(k f k i )r e ] (3.1) where e is the charge, m is the mass and k i and k f are the incident and the scattered wave vectors respectively. The quantity e 2 /mc m is the Thomson scattering length (also called the classical radius of the electron: its small value is the reason why the interaction is so weak) and the 1/R 0 represents the decay of the scattered wave with distance. Going through all the steps of the theory presented in many general textbooks [1, 3, 4] where first the interaction of a single electron, than that with an atom and finally with all the atoms in a unit cell are considered one can derive a formula for the amplitude of the wave scattered by a crystal of the form: A cryst = A 0 e 2 mc 2 1 R 0 F (q) N 1 1 N 2 1 N 3 1 n 1 =0 n 2 =0 n 3 =0 exp (i q (n 1 a 1 + n 2 a 2 + n 3 a 3 )) (3.2) where q = k f k i is the momentum transfer and in the case of elastic scattering k i = k f = 2π/λ, and N c F (q) = f i (q) exp(i q r j )d 3 r, (3.3) j=1 is the structure factor of the unit cell and N c is the number of atoms in the unit cell, defined by the vectors a 1, a 2, a 3, all atoms of the unit cell having their own atomic scattering factors f i (q). Setting now the size of the crystal equal to 0 in the direction of a 2 and a 3 it is worthwhile to study what kind of diffraction pattern one gets from a simple one dimensional crystal. In order to get an idea of how the scattered intensity would look like, one must focus on the modulus squared of the only sum remaining from equation 3.2. Disregarding for the moment the quantity F(q), we get: N S N1 (q 1 a 1 ) = exp(iq 1 a 1 n 1 ) n 1 =0 2 = 1 exp(iq 1 a 1 N 1 ) 1 exp(iq 1 a 1 ) 2 = sin2 (N 1 q 1 a 1 /2) sin 2 (q 1 a 1 /2) (3.4)

4 52 Chapter 3. X-ray diffraction on thin films and multilayers This function, called N-slit interference function (due to it s use in optics), is plotted in Figure 3.1 in the case of N 1 =10, for q 1 a 1 [ 1, 8]. This function is responsible for many of the features that we will deal with in the present chapter, such as the Scherrer formula and the intensity oscillations close to the Bragg peaks of perfect crystalline films [4]. In the limit of a very long chain of atoms (N 1 large) this function transforms into a series of delta functions centered at multiples of 2π. In reality though, the crystal is not infinite but it has to end somewhere. If one would only consider a free standing monolayer, namely N 1 =1, equation 3.4 gives S N1 =1(q 1 a 1 ) 2 = 1. This would mean that the diffraction from the monolayer will be independent of q 1 and will take on a finite, constant value whereas in the other two directions the Laue conditions are satisfied. In reciprocal space this will produce rods that are perpendicular to the surface of the monolayer. In reality the N 1 is larger than 1 and finite, therefore in this case the diffraction profile transforms into the so-called crystal truncation rod. Considering for N 1 a high, finite value, the numerator of equation 3.4 is a rapidly varying function of q 1 and therefore in a real experiment it is smeared out. In this limit the diffracted intensity will be inversely proportional to sin 2 ( q a 1 ). This means that the diffracted intensity is not 0 between the (00l) Bragg points, but [2]: 2 S( q a 1 ) 2 = exp( (i q a 1 + ɛ) j) (3.5) j=0 where ɛ is a small quantity that describes the absorption from one layer and therefore the above formula gives only significant value close to the Bragg peaks and does not diverge to where the Bragg condition is satisfied. Therefore the diffracted wave from a crystal that is built up by N 1 N 2 N 3 unit cells using the N-slit interference functions is: A cryst = A 0 e 2 mc 2 1 R 0 F (q)s N1 (q 1 a 1 )S N2 (q 2 a 2 )S N3 (q 3 a 3 ) (3.6) the diffracted intensity being the absolute square of this amplitude (I cryst = A cryst 2 ). In the limit of a large crystal there will only be nonzero diffracted intensity if the Laue conditions are satisfied, or if: q 1 a 1 = 2πh q 2 a 2 = 2πk (3.7) q 3 a 3 = 2πl

5 3.3. X-ray diffraction by crystals 53 Figure 3.1: Graphical representation of Equation 3.4 with N 1 =10. The separation of the main peaks is 2π. It is of major importance that the width of this main peak is inversely proportional with N 1, namely with the number of atoms in the one-dimensional chain and that besides the main peaks subsidiary maxima exist, the spacing of which is 2π/N 1. where h, k, l are the Miller indices. The conditions can be simultaneously satisfied if q is equal to a reciprocal lattice vector g = h g 1 + k g 2 + l g 3 where g 1 = 2π a 2 a a 3 1 ( a 2 a and similarly g 3 ) 2 and g 3, by cyclic permutation. Because epitaxial films studied here are extremely thin in comparison with the penetration depth of X- rays, besides the diffracted intensities originating from the films, an intense peak as a result of the diffraction from the substrate will always be visible. Fortunately however this substrate peak will appear usually at a different position in reciprocal space. Additionally to the crystallographic information derived from X-ray diffraction there is one other, simple, but very useful application one can profit from by the use of well focused, monocromatic X-rays, namely the so called reflectivity curve. The reflectivity measurements are possible due to the contrast in electronic density between two layers (or many in the case of multilayers) lying on top of each other. Due to the interference between the X-ray beams scattered from the different interfaces one is able to record a scan consisting of

6 54 Chapter 3. X-ray diffraction on thin films and multilayers so-called thickness fringes i.e. intensity maxima corresponding to constructive interference and minima due to destructive interference. The position of these features is directly related to the thickness of the layers. In the hard X-ray region the refractive index n of the materials is smaller than unity. This is a consequence of the repeated resonant behavior of many possible electronic transitions up to the X-ray region. The refractive index can be represented as follows: n = 1 δ + iβ (3.8) where the two real numbers δ and β, have the following material and wavelength dependencies: δ = 2πρ af 0 (0)r 0 k 2 β = µ 2k (3.9) δ is a reduction due to the binding of the electrons in the atom, which is proportional to the electron density (ρ a f 0 (0)) of the material and inversely proportional with the square of the wavevector k. The imaginary part is related to the absorbtion of X-rays, β being proportional with the absorbtion coefficient. The refractive index is directly related to the real and imaginary components of the dispersion corrections (f and f ) in the following way: n = 1 2πρ ar 0 k 2 (f 0 (0) + f + i f ) (3.10) where ρ a is the atomic density, f 0 (0) is the number of electrons in the atoms and f and f are related to each other through the Kramers-Kroning relations. Using the values for f and f tabulated in crystallographic tables [6] and calculating the overall electron density of the individual layers and the substrate from the knowledge of the lattice parameters and the atomic constituents, one can have a good starting point for the refractive index profile responsible for the shape of the reflectivity curve. The reflectivity from a slab of material can be calculated using the following equations relating the angular dependence of the reflected X-rays through the phase factor p to the layer properties such as electron density, roughness and thickness: R = r 2 = r 01 + r 12 p r 01 r 12 p 2 ( k 2 σj 2 with r ij = r ij exp sin α ) i sin α j 2

7 3.3. X-ray diffraction by crystals 55 and r ij = α i α j α i + α j (3.11) where the α i and α j are related through the generalized Snell equation and p 2 being given by: n i cos α i = n j cos α j (3.12) p 2 = exp(i2k sin α 1 ) (3.13) where represents the thickness of the layer and σ i represents the roughness parameter of the i th interface. In Figure 3.2 we show two reflectivity curve simulations of a 150 Å and 400 Å layer of CoO on a gold (upper graph) and a MgO substrate (lower graph) respectively. At small angles one can see that the reflected intensity in the case of Au substrate decreases in two steps which for the MgO substrate only one decrease is discernable. This difference clearly illustrates the effect of electron density. Gold being a heavy element, has a electronic density 2.5 times higher than the electronic density of CoO, whereas MgO, being composed of light elements, has an electronic density which is lower than the one of CoO. Making an analogy with the visible light escaping from an optically denser medium (such as water) into air there will be a critical angle under which the total reflection phenomenon inhibits the emergence of light from the water. One has the same phenomenon for the X- rays with condensed matter. Because the refractive index inside the material is now smaller than the one of air (n air = 1), there will be an angle under which the X-rays are totally reflected. This angle is called the critical angle and it is dependent on the electronic density as: 4πρel r 0 α crit = k 2 (3.14) The electronic density ρ el is the only material property appearing in this equation, thus a good fit of the reflectivity curve can give information about the stoichiometry of the layer. This is a commonly used technique in the case of novel films with unknown stoichiometry [5]. Another characteristic feature of the reflectivity curve is the so-called thickness or Kiessig fringes also pointed out in Figure 3.2. The period of these fringes gives an immediate approximate value of the thickness d, the separation of two maxima being inversely proportional with d (for small angles d = λ/2 α where α is the difference in angular position between two consecutive maxima or minima). More sophisticated methods of simulating

8 56 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.2: Simulated reflectivity curve for (top) 400Å CoO on a Au substrate. The gold having a high electronic density, its critical angle α crit is higher than the one for the CoO over-layer and therefore both can be visible. (bottom) 150 Å of CoO on MgO, where only the critical angle of the CoO is observable. The reflected intensity oscillations are called Kiessig fringes and are a consequence of the interference of X-rays reflected from the top and the bottom of the thin film the reflectivity curves give more accurate values for the thickness and give also information about the size of the inter-layer roughnesses (σ i ) or possible correlations in height fluctuations [3]. Reflectivity measurements can also be done for multilayers and the approximate formula describing the phenomenon was given by Parratt [3, 20]. In Section 3.11 an example of a measurement and a simulation is presented. To give a numerical picture for the reader, Table 3.2 shows the numbers used for the simulation of the reflectivity curve in Figure 3.2. The values for f and f were taken from [6] and are specific for Cu Kα radiation. A last important X-ray diffraction measurement method which is worth mentioning before turning to the experimental results in this chapter, is the so called Reciprocal Space Mapping (RSM). This mapping is done by transforming a series of one dimensional radial scans taken with slightly different offsets, into a 2 dimensional image. In Figure 3.3 the execution of such a RSM is shown. The radial scans are so called 2θ-ω scans (2θ is the detector angle,

9 3.4. Measurement setup 57 Parameter O Mg Co Au MgO CoO Z=f 0 (0) f f σ 1, σ Re(n) Im(n) α crit Table 3.2: Parameters used for simulation of the reflectivity curves in Figure 3.2 and ω is the angle between the sample and the incident beam), whereby only the length of the momentum transfer vector K is changed but not the direction in the reciprocal space. The omega scans are scans whereby the direction of K is changed but not the length. By combining these two motions and recording the scattered intensity as a function of these two angles (ω and θ), one can produce a 2 dimensional map of the spanned reciprocal space. The transformation of the recorded angles (2θ and ω) to K and K is done using the following simple transformations: K = K cos(δ) K = K sin(δ) (3.15) where δ = θ ω is called the offset and K=2k 0 sin(θ). Reciprocal lattice maps around non-specular Bragg peaks will prove to be useful measurements in determining the relaxation state of strained layers and they also serve as tools for understanding diffuse scattering from partially relaxed epitaxial layers. 3.4 Measurement setup The measurement setup used to collect diffraction data for the present thesis was a Philips X pert four axes diffractometer equipped with primary and secondary optics suited for epitaxial thin film analysis. In this section we will present some of the accessories that are specific for this configuration. X-rays characteristic of Copper are generated by a PW-3373/00 (Cu LFF DK ) X-ray tube operated at U acc =40 kv and I emis =40 ma in the line focus mode, the beam size being 12 mm 0.4 mm. This beam is made

10 58 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.3: Reciprocal Space Mapping with X-rays. By a controlled motion of the sample and detector one can span the shaded area with the momentum transfer vector ( K), recording the diffracted intensity. The collected data are plotted in a 2D graph where the color of every pixel will represent the diffracted intensity. Usually the horizontal axis will correspond to the K and the vertical axis to the K directions. parallel (horizontal divergence 0.05 ) and monochromatic by the hybrid X- ray mirror sketched in Figure 3.4. The spectral range of this monocromator is larger than the so-called four-crystal monochromator but delivers a much more intense beam. Because the K β component is almost completely suppressed by the monochromator, the remaining beam is monochromatic composed mainly of K α1 (λ= Å) the K α2 radiation being suppressed to less than 0.1 %. The divergence slit commonly used was the 1/8 fixed slit which produced a beam of mm after the monochromator a vertically divergent and horizontally parallel beam. After diffraction or reflection from the sample follows the secondary optics. Here the antiscatter slit is meant to reduce the amount of scattered X-rays by other objects than the sample itself (usually

11 3.4. Measurement setup 59 Figure 3.4: Picture of the inside of the Philips X pert diffractometer. The X-ray source is fixed, the sample can be rotated around the vertical (ω), horizontal (ψ) and around the azimuth (φ) axes. The detector at the left of the picture rotates around the vertical axis positioned at the center of the sample (2θ). Also presented is a schematic drawing of the hybrid monochromator if the mirror is viewed from above. taken to be 1/4 ). The next important element is the 0.04 rad Soller slit consisting of horizontally closely spaced metal foils, which reduces the vertical divergence of the scattered beam to some extent. Although it reduces the intensity by a factor of 2, the noise level is also drastically reduced. The last element positioned before the detector is the Programmable Receiving Slit, which is a slit that can be controlled by software and is meant to define the resolution of the apparatus (since the current configuration contained no analyzer crystals). The detector used to count the number of photons at a certain detector angle (2θ) is a proportional detector, fabricated as a sealed chamber filled with a xenon/methane gas mixture. These detectors have a 99 % linearity range of 0-500,000 cts/s and a 84 % efficiency for Cu radiation [8, 9].

12 60 Chapter 3. X-ray diffraction on thin films and multilayers 3.5 Strain relaxation in epitaxial layers Before entering into the details of our experimental findings on the relaxation of epitaxial transition metal oxides on different substrates and buffer layers it seems appropriate to first give a flavor of more than 50 years of research committed to reveal the intricate mechanisms through which homogeneous layers relax the epitaxial strain imposed by the substrate. Obviously, since this thesis is not devoted to conceive a new, more precise theoretical description of relaxation, we will just present what we think are the most important details in some of the easily comprehensible theories developed throughout the years. This research area developed so rapidly mostly because of the advances in semiconductor industry in the direction of manipulation of electronic properties of thin films deposited on mismatched substrates. These developments required quantitative theories relating values for the strain in these films with known quantities such as thickness, mismatch and elastic constants. The theoretical research started in year 1949 with the work of F.C. Frank and J.H. van der Merwe [10] who developed their original equilibrium theories which showed that only below a certain critical thickness the film will accommodate the mismatch despite the strain in the film (see also [12]). Layers having thicknesses above this critical value relieve their strain by formation of misfit dislocations. The theory named after their discoverers Frank-van der Merwe usually gives an underestimate of the critical thickness. Matthews and Blackeslee [11] came up with a more realistic formula for the thickness dependent strain based on the energy minimization of the sum of the elastic energy of the film and the energy of an edge dislocation. This formula is also designated after their names (M-B model): ɛ(h) = G s b (G s + G f ) 4π(1 + ν)h [ln(h ) + 1], (3.16) b for layers having the dislocation in the interface plane, and ɛ(h) = (1 ν) 2G f h cos(λ)(1 + ν) (3.17) [ Gf G s b(1 ν cos 2 (α)) [ln( 2π(G f + G s )(1 ν) hb ] ) σ sin(α) + γh ] b cos(φ) assuming the more general case of dislocations, not lying in the interface plane,

13 3.5. Strain relaxation in epitaxial layers 61 Figure 3.5: Determination of critical thickness from Equation 3.17 as explained in the text. For some cases the curve does not cross the y=0 line and therefore, the mismatch is too high for any nonzero critical thickness. In this case dislocation are supposed to appear instantaneously in the first monolayer. This is in reality not true as explained further in the text, because for very thin films one must redefine the core energy of the dislocation. where ɛ(h) is the thickness dependent strain, G s and G f are the shear modulus for the substrate and the film respectively, α is the angle between the dislocation line and its Burgers vector, λ is the angle between the slip direction and the line in the interface plane which is normal to the line of the intersection between the slip plane and the interface, and φ is the angle between the normal to the slip plane and the sample surface, σ is the surface tension of the film and γ the stacking fault energy. We plotted the results of this prediction (eq. 3.17) in Figure 3.5 for most of the films discussed in this chapter. The critical thickness is the second intersection of the curve with the y=0 axis (pointed at by the arrows). We plotted the function (ɛ(h) f 0 ) where f 0 is the coherent mismatch defined by Equation 2.5 in the previous section f 0 = (a film a substrate )/a substrate. It is important to note that while the CoO is predicted to relax at 55 Å on an MgO substrate, it relaxes after only a few monolayers on MnO and that the mismatch is so large that one can no longer define a critical thickness for CoO

14 62 Chapter 3. X-ray diffraction on thin films and multilayers on Ag and MgAl 2 O 4 1. Table 3.3 contains the material parameters used for the calculation of critical thickness displayed in Figure 3.5. Material Lattice parameter (Å) ν G (N/m 2 ) MgO Ag MgAl 2 O MnO CoO Table 3.3: Parameters used for Figure 3.5. Theory which takes into account also dislocation multiplication and therefore includes also the temperature and time as an important factor is the model of Dodson and Tsao [13 15]. Their model was a breakthrough towards understanding the reasons why the usual experimental data showed an increased value of the critical thickness compared to the values given by the Frank-van der Merwe and Matthews-Blackeslee (M-B) models. They propose a differential equation describing the strain relief γ(t) of the form: dγ(t) dt = CG 2 [f 0 γ(t) ɛ(h)] 2 [γ(t) γ 0 ] (3.18) here f 0 is the coherent mismatch, γ 0 is a dislocation source term and ɛ(h) is the residual strain at equilibrium given by: ɛ(h) = 1 4π (1 ν) cos 2 (α) ln( ξh b ) 1 + ν ( h b ) (3.19) with ξ [1, 8] being a parameter accounting for the core energy of the dislocation. This model predicts a more sluggish relaxation of the misfit strain. Finally we want draw your attention to a more recent observation of K. Wiesauer and G. Springholtz [16] who pointed out that the factor ξ, which is usually taken as a constant when determining the strain with the M-B model, should in fact be considered to be thickness dependent. They also pointed out that ξ accounts for the energy of the dislocation at the dislocation core and For the angles in Equation 3.17 we used α = π/2, λ = π/4, φ = π/4, specific for the edge dislocations occurring in ionic solids and σ = γ = 0 thus neglecting surface tension and staking faults.

15 3.6. X-ray specular scans from CoO on MgO 63 Figure 3.6: Dislocation system in the case of (a) diamond structures b= 1 2 <1 10>{111} and in the case of (b) rock-salt structure b= 1 2 <1 10>{110}. The white arrows represent the Burgers vectors and the dark shaded areas are the slip planes. at very low thicknesses this parameter should have different values which also depend on the material. Based on this uncertainty they treated ξ [1, 8] as an adjustable parameter and succeed to fit quite well the strain relaxation of PbTe 1 x Se x layers on PbSe (001). As a closure of this section we must mention that the kinds of misfit dislocations which relieve the strain are different for oxides and semiconductors (there are very few studies dealing with the relaxation behavior of epitaxial strain in thin films of binary oxides). To illustrate the difference, the usual dislocation system in semiconductors and in rock-salt oxides are compared in Figure 3.6. In covalent solids such as Si and Ge with strong directional bonds and having a diamond structure [17], the strain is relieved by the appearance of dislocations of the type b= 1 2 <1 10>, with slip planes {111} as shown in the figure. For binary oxides the relaxation occurs though the b= 1 2 <1 10> with the {110} slip planes [17]. Because the dislocation is quite different in the two structures, the displacement field, by which the atoms of the structure rearrange, will also be different. This will become clear in the sections concerned with diffuse scattering where we will show that the two dislocation types produce a different diffuse intensity profile. 3.6 X-ray specular scans from CoO on MgO Tn this chapter we shall refer to scans having offsets in the range of δ ( 1, +1 ) (see Figure 3.3), these are the so-called specular scans. In fact the miscut, i.e. the angle between the surface normal and the (001) crystallo-

16 64 Chapter 3. X-ray diffraction on thin films and multilayers graphic direction of the substrate is responsible for the often nonzero values of δ. This miscut appears as an error in the polishing direction in the case of epipolished substrates and as an unavoidable effect when the crystals are cleaved. Specular scans produced by scanning the size of the momentum transfer perpendicular to the substrate give information about the lattice spacing of the film and the substrate perpendicular to the surface. Figure 3.7 presents two such specular scans for two different CoO layer thicknesses epitaxially grown on MgO cleaved substrates. Both films were covered by a thin ( 10 Å) MgO layer to avoid oxidation of the surface. In both figures two groups of peaks are visible and they correspond to the (002) (at θ 21 ) and (004) (at θ 46.5 ) reflections of the MgO and CoO FCC lattice. The absence of any other peaks than the (002) and (004) is a proof (besides the RHEED results) that the films are perfectly epitaxial. The position and the width of the peaks gives information about the c( a=b) value of the lattice constant and the thickness of the film. In the inserts of the upper figure intensity oscillations are observable, they appear as a consequence of the limited thickness of the film ( 150 Å CoO on MgO) as debated in section 3.3, the subsidiary maxima originating from the N-slit function (see Figure 3.1). In the bottom figure which is a specular scan of a 450 Å CoO film, these oscillations are only vaguely visible. As we will demonstrate later in this chapter, these oscillations disappear because of the inhomogeneities of the out-of plane lattice constant, caused by the multiplication of misfit dislocations in the film. In Figure 3.7 simulations of the scans are also presented at the bottom of the graphs as well as in the inserts (gray lines). These kinematic calculations take into account not only the complex refractive indices of the substrate and the film (see section 3.3) but also the angular dependent scattering factors of the atomic components [6, 7]. A reasonably good fit can be achieved using this kinematical approach in the case of thin ( Å) CoO films, but as one can observe in the insert of the bottom figure, these simulations do not fit the measured data at large thicknesses. This limitation is mainly due to the multiplication of the above mentioned inhomogeneities in the c lattice parameter at higher film thicknesses. Another effect may be noticed if one compares the width of the peaks in the simulation with the experimental curve. In the case of the thin (150 Å CoO) film the width of the two are equal but in the case of the thicker (450 Å CoO) films the experimental curve has a wider (002) Bragg peak than the simulation. This is again a consequence of the inhomogeneities of the c lattice parameter and the broadening will be explained by means of diffuse scattering from networks of misfit dislocations.

17 3.6. X-ray specular scans from CoO on MgO 65 Figure 3.7: Specular scans from (a) 150 Å and (b) 450 Å CoO films on MgO (001) together with the simulation (below) using a kinematical approach. The inserts show a magnification of the (002) and (004) Bragg peaks. The thin film can be almost perfectly fitted with the simple approach but the thick film has broader Bragg peaks than the simulation and the expected thickness fringes are mostly vanished.

18 66 Chapter 3. X-ray diffraction on thin films and multilayers In the following we show how the out-of-plane lattice parameter c, determined from these types of specular scans, depends on the CoO layer thickness. Figure 3.8 shows this dependence for films having increasing thicknesses. The dashed line represents the bulk value of the lattice constant for CoO, namely Å. Because of the compressive strain of the substrate this value is not reached for the thickest films we prepared (860 Å). Therefore up to a value of 350 Å the c parameter stays constant at a value of Å. Knowing the c parameter and using the result (discussed later) that the in-plane lattice parameter of CoO coincides with a MgO =4.212 Å (coherent growth) one can determine the Poisson ratio of CoO, defined as the negative ratio between the lateral and longitudinal strain under uniaxial longitudinal stress. In our case of biaxial stress: ɛ = 2 ν 1 ν ɛ (3.20) where ɛ = (c a CoObulk )/a CoObulk and ɛ = (a MgO a CoObulk )/ /a CoObulk , From this we determine the Poisson ratio for CoO: ν CoO The out-of-plane lattice constant c remains Å up to a thickness of Å, plastic relaxation occurs and the reduction of strain evolves through the appearance of suitable dislocations at the layer substrate interface. This will influence not only the the position of the peaks but also their shape. The Figure 3.8 (b) shows the width of the (002) Bragg reflections measured from the specular scans of CoO films grown on MgO having increasing thickness. As explained in section 3.3 the width of the peaks should be inversely proportional with the number of monolayers. This effect is used also in interpreting the X-ray diffraction spectra from polycrystals to determine the grain size and is given by the Scherrer formula: w = k λ d cos(θ Bragg ) (3.21) where w is the full width half maximum (in θ) of the peak, d is the thickness of the film and k is called the shape factor, in the present case of thin films taking a value of k = 0.88 [3]. Therefore, not only from the position in 2θ Bragg but also from the width of these peaks results that the critical thickness of CoO on MgO (having a misfit of f ) is 350 Å. This conclusion is drawn only on the basis of the specular scans described in this section. In the following we show that off-specular scans give a different value for this critical thickness.

19 3.6. X-ray specular scans from CoO on MgO 67 Figure 3.8: (a) The points represent the thickness dependence of the out-ofplane c lattice constant measured by means of specular scans from the position of the (002) Bragg reflection. The thick solid line gives the lattice parameter evolution as predicted by the M-B formula from Section 3.5 and the dashed line represents the bulk lattice parameter of CoO. (b) Peak width of the (002) reflections from the specular scans as a function of layer thickness (points). The drawn line is a fit using the Scherrer equation. At thicknesses higher then 360 Å, the widths are considerably larger than expected, the cause being the diffuse scattering produced by the dislocations.

20 68 Chapter 3. X-ray diffraction on thin films and multilayers 3.7 Reciprocal space maps from CoO on MgO In the following we will be concerned with the type of scans described in detail in Section 3.3. These Reciprocal Space Maps (RSM) can be produced around specular or non-specular reflections. In the present section we will only present off-specular RSM s and deal in more detail with the specular RSM s in the following section. In the case of coherent growth, the in-plane lattice parameter of the film will be coincident with the one from the substrate, whereas the out-of-plane lattice parameter will increase according to equation In reciprocal space this is equivalent with having the in-plane Bragg reflections of the film at the same position as the substrate but at a displaced position in the out-of-plane direction. Figure 3.9 shows a schematic picture of the off-specular RSM s for the case of CoO on MgO. From the positions of the intensities in reciprocal space one can calculate both the out-of-plane and in-plane lattice parameters. Figures 3.10 and 3.11 show RSM s around the (224) reflection of CoO and the MgO substrate measured from films of different thicknesses. The axes in these maps are in so-called Reciprocal Lattice Units (RLU) which are reciprocal space units normalized to the theoretically achievable maximum distance from the origin for a certain X-ray wavelength. The normalization constant is equal to the diameter of the Ewald sphere (2k 0 =2 (2π)/λ). At low thicknesses (125 Å) the growth seems to be coherent, and the off-specular Bragg reflection is situated straight below the substrate reflection as expected from the previous reasoning. This is not the case anymore for thicker films and starting from a thicknesses of 170 Å of CoO, there appears to be extra intensity extending outside this value in the k direction as can be seen in the bottom RSM in Figure 3.10 (b). Although from the data presented in Figure 3.8 we deduced that the growth remains coherent up to Å, this RSM shows evidence for the contrary. Therefore, the two measurements i.e. specular scans and RSM around off-specular Bragg peaks seem to give inconsistent results. These differences substantiate that Reciprocal Space Mapping is a more sensitive tool for detecting the appearance of the relaxation of thin films, than the simple specular scans. Figure 3.11 presents RSMs of thicker samples. Here one observes that with increasing thickness the maxima of the intensity distribution moves away from the homogeneously strained position towards the relaxed position situated at k =0.722, k = for relaxed CoO in units of 2k 0. This intensity distribution is produced by the dislocations which multiply with increasing thickness [18 22, 24, 31] and we call it the dislocation induced diffuse scattering. One would obviously like to quantify, in some manner, the peak position

21 3.7. Reciprocal space maps from CoO on MgO 69 Figure 3.9: Schematic representation of the RSM for off-specular reflections of CoO on MgO. Due to the compressive strain, the c parameter of the CoO will increase, which corresponds to a decrease in the vertical direction in the reciprocal space. The unrelaxed CoO would have its (hkl) reflection straight below the MgO reflections, whereas the relaxed CoO would produce the diffracted intensity to be situated along the X-ray momentum transfer K. The surrounding of the (hkl) reflections of the MgO and CoO is the RSM described in section 3.3. and the intensity distributions of these reflections. One way of doing so is to select the intensity from the rectangular area displayed in Figure 3.11 and to determine the position of the maxima (k, k ) of the integrated intensity in

22 70 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.10: Area scans around the (224) Bragg reflections of CoO and the substrate MgO measured for samples of different CoO thicknesses. As explained in the text and schematically in Figure 3.9, the thin films which grow homogeneously on the substrate will have their Bragg reflection straight below the substrate peak. In the upper picture an example of this behavior is shown (125 Å CoO on MgO). At a thickness of 170 Å (b) the Bragg reflection of the film moves towards the relaxed bulk value. The intensity scale (the darkness of the pixels) is logarithmic because of the high intensity ratio between the scattering from the substrate and overlayers.

23 3.7. Reciprocal space maps from CoO on MgO 71 Figure 3.11: Like the previous picture but with higher thicknesses of the films. According to the specular scans 3.8 (a) the layer responsible for the upper graph is still in the unrelaxed state, which is obviously not the case from this figure. The bottom figure is a RSM from the thickest CoO film grown on MgO. The rectangular box around the CoO (224) reflection is the area from which we determined the lattice parameters as explained in the text.

24 72 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.12: Lattice parameters measured from the position of the maxima of the integrated intensity along the k and k of the CoO (224) reflections on MgO. The data was corrected by shifting the values such that the MgO (224) is situated at it s theoretical position. This was necessary due to the occasional misalignment caused by the surface miscut. the directions indicated by the arrows. The lattice parameters obtained in this way are shown in Figure Although the values of the out-of-plane lattice parameter are much more scattered, they compare well with the values already discussed in Section 3.6, Figure 3.8 (a), suggesting the same critical thickness. Therefore, although visible, the diffuse intensity around these reciprocal lattice points is not intense enough to change the position of the maxima originating from the coherently strained lattice. However, since the critical thickness is defined as the thickness at which dislocations start to appear, the thickness at which this diffuse scattering appears around these off-specular spots represents the true critical thickness that can be determined with this technique. We therefore can reevaluate our experimental critical thickness for CoO on MgO of being round 170 Å. As a short preliminary conclusion of this section one can say that although we could observe the diffuse scattering produced by the dislocations in the film, this intensity was not high enough to displace the measured maximum position. Therefore, the thickness at which diffuse scattering appears, is defined as being the borderline between homogeneous and relaxing regions. As a consequence, two obvious questions arise, namely: 1. Will the diffuse intensity around Bragg peaks having higher intensity such as the (002) also be higher and therefore more easily observable? 2. Is it possible to simulate the diffuse scattering due to dislocations?

25 3.8. Diffuse scattering around the (002) reflection of CoO on MgO 73 The answers to these questions will be given in the following sections. 3.8 Diffuse scattering around the (002) reflection of CoO on MgO In this section we shall consider the first point mentioned at the end of the previous section and we will therefore search for diffuse intensity around Bragg spots having higher intensity such as the (002) reflections. These measurements can easily be done by performing RSM s around these areas in the reciprocal space. Results are shown in Figures 3.13, 3.14 and 3.15 for samples of increasing CoO thickness. As in Figures 3.10 and 3.11 the axes are in units of the Ewald sphere diameter (2k 0 ) but to the right of the RSM s also another scale is drawn, namely (q d), a scale by which all the figures can be compared. Such a scale will become useful in the following section when we deal with the simulations of diffuse scattering. These maps contain three important regions: the MgO substrate (002) reflection situated around (k = 0.365, k = 0), the CoO film (002) Bragg reflection at (k = 0.358, k = 0) which is very sharp in the k direction but elongated in the k direction, and finally the diffuse scattering situated to the left and to the right of the film (002) Bragg peak, which is not very intense for the thinnest sample, but clearly visible for thicker samples. To the left of every RSM two intensity profiles are plotted. They represent the integrated intensity along the k from the boxes surrounded by the rectangles in Figure 3.13 (a): The continuous line corresponds to the intensity profile of the coherent part of the scattering, the broken line corresponds to the diffuse part of the scattering. The intensity scale is set to logarithmic since the intensity of the substrate peak is orders of magnitude higher than the intensity scattered by the film. There are several features one observes. Firstly, the line scan represented by the solid line corresponding to every RSM is equal to a specular scan as discussed in Section 3.6 (see also Figure 3.7). The relaxation of the c lattice parameter of Figure 3.8 (a) can therefore be determined from such kind of RSM s. Secondly, the scans originating from the diffuse scattering (dashed box and lines) have their maxima displaced with respect to the (002) peaks originating from coherent scattering. The maxima are located at higher k value which imply a lower, relaxed, c value. Thirdly, weak diffuse intensity is already present for the thinnest (130 Å) sample Figure 3.13 (a). Therefore dislocation formation already has started at this low thickness. It is thus true that the diffuse scattering expected around higher intensity peaks is more pronounced. Defining as before the true critical thickness as

26 74 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.13: Reciprocal Space Maps around the (002) reflections of the MgO substrate and the CoO film having increasing thicknesses. The axis on the left side of the RSM s are in units of Ewald sphere diameter (4π/λ) and the axis on the right side is in (q x d), i.e. dimensionless units centered on the CoO (002) peak. On the left-hand side of every RSM are graphs presenting integrated intensity scans corresponding to the boxes depicted in the area scan. Graphs on the left side correspond to selections from the boxes exemplified in the upper figure with corresponding line types.

27 3.8. Diffuse scattering around the (002) reflection of CoO on MgO 75 Figure 3.14: Diffuse intensity appears already at films of 130 Å thickness (at k 0). The maximum of this diffuse intensity (dashed line) is shifted to higher k values in comparison with the coherent scattering situated at k =0 (solid line). This situation corresponds to the low density limit (see text in next section). The boxes in the RSM in figure (b) are selections wherefrom the diffuse (wide box) and coherent (narrow box) intensity was determined presented later in this Chapter in Figure 3.22.

28 76 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.15: At very high thicknesses the specular peak position (collected from k =0 (solid line)) will move more towards the diffuse intensity maxima (collected from k 0, dashed line). This corresponds already to the intermediate-to-high density limit (see text).

29 3.9. Modelling of the diffuse scattering 77 the value at which diffuse scattering appears for a film-substrate system the relaxation for CoO grown on MgO already sets in at 130 Å. In the following we will try to numerically simulate the observed diffuse scattering around specular and of-specular reflections. 3.9 Modelling of the diffuse scattering In this section we will focus our attention on the discussion of the main hypothesis and predictions of some models explaining the appearance of diffuse scattering from epitaxial layers. The simplest model was envisaged by V. Srikant et al. [25] and was meant to explain in a very simple manner the relaxation of epitaxial layers having a large lattice mismatch. The width of the (002) or (004) specular reflection (the omega scan) perpendicular to the crystal truncation rod was for long recognized as a quality factor for epitaxial films. It was interpreted by Srikant and co-workers as mosaic spread of the films, namely as a mis-orientation of the grains and sub-grains of the film with respect to each other and to the substrate. This mosaicity was described by two parameters: the tilt range and the twist range (expressed as the full width half maximum (FWHM)) of the film with respect to the sample normal. Others such as Chierchia and co-workers [26] used four parameters in their analysis, the lateral and vertical coherence length being the other two in addition to the ones of Srikant et al. The mosaic of the films is explained by the appearance of, what they call, geometrically necessary misfit dislocations to fill the space and to account for the sub-grain mis-orientation distributions. The next model worth mentioning in the present context is the one developed by P. Kidd, N.L. Andrew and P.F. Fewster [27, 31]. They studied RSM s around the (004) specular reflection from different thickness of In 0.1 Ga 0.9 As layers on [001] oriented GaAs substrates. The layers having this composition produced a small misfit between the substrate and the film lattice parameters of 0.63 %. They explained their observations, which were very similar with our RSM s taken from films of CoO on MgO substrates, by supposing that the dislocations appearing at the interface, after the start of the relaxation, produce a lateral region having a different lattice parameter than the rest of the strained film (see Figure 3.16(a)). These regions around the dislocation lines are extending to distances that is proportional to the thickness of the film (TEM observations). These regions give also rise to the diffuse intensity next to the strained Bragg peaks, as they were able to prove by the use of X-ray topography technique [5, 28 30]. In their model, the diffuse peaks are

30 78 Chapter 3. X-ray diffraction on thin films and multilayers interpreted as the square of the Fourier transform of the constant strain field having a finite lateral extent proportional to the film thickness centered on the dislocation line. This approach was to some extent successfully used by Großmann et al. [32] (Figure 3.16(b)) and H.R. Relß et al. [33]. It is evident from the data of Großmann et al. (reproduced here for the sake of clarity) that the diffuse peaks can be properly fitted by the above described model but as the dislocation density increases due to annealing, the model does not provide a good description anymore. The schematic representation of this model is presented in Figure 3.16 (a) together with the above mentioned Fourier transform : I diffuse (k, s) [sin(k s/2)/(k s/2)] 2, s being the lateral extent of the strain field and k the parallel momentum transfer of the X-rays. In addition to the fact that the model cannot fit the high density limit, it cannot properly describe the RSM s around non-specular reflections either. More elaborate and more realistic models describing the diffuse scattering were developed independently by Kaganer [21] et al. and Holý et al. [18 20]. Their approach is virtually the same except for some details. Below we shall give a short description summarizing Kaganer s model which we consider to be the most valid one of the two, as we shall discuss at the end of this section. Kaganer took into account the total displacement field of the strain relieving misfit-dislocations which are expected for a certain crystal structure (see section 3.5 Figure 3.6). The necessity to do this becomes obvious if one observes the displacement field, noted as u s, originating from a 45 tilted edge dislocation (referred to as normal type of dislocation for rock-salt from here on) plotted in Figure 3.17 as given in the appendix of ref [21] and first solved by A.K. Head [38]. It is also clear from this picture that the displacement field is only appreciable close to the dislocation. The extent is indeed comparable to the thickness of the film. It is also obvious, however, that the approximation of a constant displacement in the most affected region is a very crude (and unnecessary) oversimplification of reality. The intensity scattered by such a film is calculated using the kinematical approximation taking into account a random distribution of dislocations starting at the substrate layer interface. It is given by I(Q) = A(Q) 2 averaged over the random positions of the defects, where A(Q) = exp[iq(r s + u s )] s is the amplitude of the wave and the sum is taken over all the atoms of the crystal. By writing it out in more detail the intensity becomes: I(Q) = s,s exp[iq (R s R s )] G(R s, R s ) (3.22)

31 3.9. Modelling of the diffuse scattering 79 Figure 3.16: (a) Schematic drawing of the Kidd-Andrew-Fewster (KAF) model. The dark-gray areas are dislocation affected regions having a lattice constant which is slightly different from the one of the light-gray areas which are assumed to be homogeneously grown. The graph shows a representation of the square of the Fourier transform from such a step function with limited width s=10, plotted against k (the X-ray momentum transfer parallel to the surface). (b) Experimental (solid line) and fitted (dashed line) is an ω scan taken from reference [32] of an as grown ZnSe layer on GaAs (upper graph) and the same for an annealed film (lower graph). The KAF model seems to fit very well the diffraction pattern before heat treatment but not after, in other words at high dislocation densities the model does not give satisfactory results. where G is the correlation function: G(R s, R s ) = exp[iq (u s u s )] = exp[t (R s R s )] (3.23) Depending on the density of defects (dislocations), one can distinguish two limits: the spatially independent (low density limit) and the spatially correlated case. In the first case the defects are so rare that they do not interact with

32 80 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.17: Displacement field due to a normal type edge dislocation. The horizontal line represents the borderline between the substrate (top) and film (below the line). The numbers represent atomic positions parallel (horizontal) and perpendicular (vertical) to the surface. The displacement is represented as brightness of the pixels, the maximum displacement being equal to one atomic distance (displayed as white), i.e. one lattice constant, the entirely black pixels indicating no displacement at all. (After the formulas given by Kaganer et al. [21].) each other and the only parameter is their density c 1. This is the limit we are considering in this section. In this case, after some approximations valid in this limit, one can write: T (R s, R s ) = c t {1 exp[iq(u st u s t)]} (3.24) where, u s,t = u(r s R t ) is the displacement at the position R s due to a dislocation at position R t. T (R s, R s ) is also sometimes referred to as the correlation function. Equation 3.24 is the correlation function used in this section. The case of spatially correlated defects has to be considered when dis-

33 3.9. Modelling of the diffuse scattering 81 locations are so close that they are no longer independent from each other. Therefore, in order to minimize the energy they tend to redistribute. At these high densities, dislocations are almost equidistant, for further examples see references [16, 34, 35]. Returning to the Kaganer model, and assuming spatially uncorrelated dislocations we distinguish two limits: the low and high density limit. In the general case the diffuse intensity can, as presented in Kaganer s paper, be calculated by numerically evaluating the following expression: I diffuse (q x, q z ) = + dx d d 0 0 dz s dz s e q xx+iq z (z s z s ) {e T (x,z z,z s ) e T (z s,z s ) } (3.25) where (q x, q z ) is a vector around a certain reciprocal lattice point Q, and T (z s, z s ) = lim x T x (z s, z s ) is the non-vanishing limit of the correlation function. This diffuse intensity is superimposed on the coherent intensity d I coherent (q x, q z ) = 2πδ(q x ) 0 d 0 dz s dz s e qz(zs z s ) e T (zs,z s ) (3.26) i.e. the sum of the two is the total intensity. The diffuse intensity here was calculated by performing a Fast Fourier transform, corresponding to the integral over dx, of the following two integrals over dz and dz evaluated as simple sums in equation In the low density limit the dimensionless parameter ρ d is smaller than unity. Since dislocations are very distant, the correlation function G(r 1, r 2 ) vanishes for (r 2 r 1 ). The Bragg reflection will still be visible but decreases in intensity with increasing density and thickness. A surprising result is that the Bragg peak does not follow the mean distortion. This is so, because the displacement field which is large close to the dislocation line, contributes largely to the mean distortion (Q u(r)) but since the displacements are comparable to the Burgers vector (which is by definition a lattice vector) they do not change much the Bragg position (sin(q u(r))). In this case the correlation function T(x, z s, z s ) is small and one can use the expansion for the exponential function exp(t) 1-T in equation 3.25, and the diffuse intensity becomes:

34 82 Chapter 3. X-ray diffraction on thin films and multilayers I diffuse (q x, q z ) = α ρ α d dx 0 dze iqxx+iqzz [e iquα(x,z) 1] 2 (3.27) This is the expression we used in calculating the low density limit (ρ d 1). The integral over dz was calculated as a sum while the integral over dx was performed by a Fourier transform of the sum. The results of these calculations will be presented in the next section. The other limiting case is the high dislocation density limit which corresponds to the case where only points close to each other are correlated and contribute to G. Here the mean distortion determines the position of the diffraction peak. The phase factor is Q (u(r 2 ) u(r 1 )) (r 2 r 1 ) (Q u(r)). Consequently the peak position depends on (Q u(r)) and the peak width depends on the mean square distortion [ (Q u(r))] 2. In this situation the coherent Bragg intensity is exponentially small and the diffuse intensity is given, in the simplest case of specular reflections, by: where d dz I(q x, q z ) = π exp[ q2 x q2 z ], (3.28) wx w z 4w x 4w z 0 w x (z) = ρ 2 Q2 z w x (z) = ρ Q 2 z + + dx[u (x)2 z,x dx[u (x)2 z,z + u (z)2 z,x ], + u (z)2 z,z ], (3.29) using u i,j = u i x j and (i, j) = (x, z). u(x,z) is the above mentioned displacement field and the symbols in the parentheses represent which type of Burger s vector is responsible for this displacement field ((x) for b x, (y) for b y and (z) for b z ). The case of off-specular reflections is slightly more complicated: d dz I(q x, q z ) = π exp[ 1 detŵ 4 w 1 ij (q q 0) i (q q 0 ) j ], (3.30) 0

35 3.9. Modelling of the diffuse scattering 83 where w ij is a matrix having elements given by: w xx (z) = ρ 2 + σ=x,y w xz (z) = ρ 2 Q xq z w zz (z) = ρ { 2 dx(q 2 xu (σ)2 x,x + σ=x,y + σ=x,y + Q 2 zu (σ)2 z,x ), dx(u (σ) x,xu (σ) z,z + u (σ) x,zu (σ) z,x), (3.31) dx(q 2 xu (σ)2 x,z + 2Q 2 zu (σ)2 z,z ) + + dxq 2 xu (y)2 x,z A more insightful description of the theory can be found in reference [21]. We numerically simulated RSM s and one dimensional scans using these two limits with the above presented formulas. Calculations using the model of Kaganer are very few [36, 37] although its results, as will be visible in the following sections, are surprisingly good, and the only fitting parameter is (ρ d). Of course one has to know in advance what type of dislocations can be present in the layer. The type of dislocation can also be identified by comparing the simulations with the experiment. It is convenient to plot the numerical calculations as a function of the dimensionless parameter q d. For the purpose of comparison between theory and experiment these axes were also drawn for RSMs in Figures 3.10, 3.11, 3.13, 3.14 and As mentioned before, Holý et al. developed also a model to describe the diffuse scattering in mismatched epitaxial layers. There is, however, a fundamental difference between the initial assumptions of the two theories. Holý starts by assuming that the differences (u(r 2 ) u(r 1 )) consist of displacements u(r i ) behaving as Gaussian random variables, Kaganer makes use of the results of Krivoglaz based on Poisson statistics for uncorrelated dislocations [39]. The simplified consideration of Holý is based on the ease with which one can arrive at a simplified form of the correlation function using the Baker-Hausdorff theorem (see appendix of ref. [3]) < exp[iq(u(r 2 ) u(r 1 ))] >= exp{ 1 2 < [iq(u(r 2) u(r 1 ))] >} or for further details see reference [19]. We believe that Kaganers treatment is the correct one. However, probably because of the rather involved mathematics it is not much used in the literature. Finally we want to mention a number of LEED studies of diffuse scattering i.e. the one by M. Dynna et al. [22] as well as the one by M. Klaua et al. }

36 84 Chapter 3. X-ray diffraction on thin films and multilayers [23]. In simulating the spot profile of thin MgO films on Fe(001) surface they also considered the full displacement field u(r). Because of the surface sensitivity of the LEED technique only the displacement of the surface has to be taken into account. Their calculations were done in the low density limit neglecting the correlation of the dislocations although the misfit of the films with the substrate was high. Nevertheless, their calculations fitted quite well the experimental profile Fitting the experimental diffuse scattering data In the previous section we have presented some of the models which were developed to explain and simulate the experimentally measurable diffuse scattering originating from misfit dislocations. In the following two sections we will use the method of Kaganer described in some detail in the previous section and apply this model to our measured diffuse scattering. We will separate the very low to intermediate dislocation density case from the high density case since these are the two regions applicable to the thin CoO layers on MgO and the thick MnO buffer layers on MgO respectively Low and intermediate density limit We have already shown RSM s around the (002) reflection of CoO layers of increasing thickness on MgO. In the following we will turn our attention to specific cross-sections from these area scans, namely scans parallel to the surface (so-called q x scans). The scans measured for CoO layers on MgO are plotted in Figure 3.18 and, as mentioned before, the horizontal axis is plotted in units of (q x d) so that all the scans can pe represented by the same axis. In this figure there are roughly 4 distinct regions visible. 1. The first region is represented by the 115 Å CoO on MgO and smaller thicknesses (not shown). Apart from one single sharp peak at q x d=0 which is entirely due to coherent scattering, almost no diffuse scattering is visible. 2. The second region is comprises the Å thick CoO films on MgO where a separate, distinct feature is visible at both sides of the, still present, coherent peak. One should note that as the thickness increases, the coherent peak is becoming smaller in comparison to the diffuse lobes. Also interesting to note is that the side lobes stay at the same position in the q x d plot as the thickness of the film is increasing and that the center coherent peak remains sharp at higher thicknesses.

37 3.10. Fitting the experimental diffuse scattering data The third region is limited to the Å thick films of CoO on MgO. Although the center peak is still present, the diffuse features have a different shape than the previous region. These diffuse lobes now become more smeared out invading the middle coherent peak and extending to higher values on the q x d scale. 4. Finally the last region is made up by the very thick films (880 Å and thicker) of CoO on MgO. The peak is entirely diffuse in character although according to some authors this is because the grain and subgrain mis-orientations. This limit will be discussed in more detail in the following subsection In Figure 3.19 we present the best fits we could achieve using the procedure outlined in Section 3.9 with the Equations 3.25 and Here we made the approximation that the values of q x around a certain reciprocal lattice point (Q x, Q y, Q z ) are so small that the integral over dx could be replaced by the Fast Fourier Transform of the remaining two integrals over dz and dz, which were calculated by evaluating the integrals as finite sums. The fits are not perfect but they mimic quite well the main features of the experimental scans. As can be seen in Figure 3.19 the (002) reflections are surrounded by one satellite on both sides of the peak while the (004) peaks have two such satellite peaks. To our knowledge these scans and simulations are unique in the case of epitaxial oxides. In many studies of thin oxide films, the width of the reflections in the q x direction is considered a measure of the quality of the films, sharper peaks representing better quality. Therefore, in the investigations published in the literature the films are either completely coherent so the q x scans are extremely sharp (and are determined by the resolution of the X-ray apparatus), or the large widening of the peak is considered as an effect of grain or sub-grain misorientations due to misfit. As disclosed in the next section, this broadening can be treated in the high dislocation density limit. It is also interesting to note that we did not use analyzer crystals in front of the detector. Our resolution was around of Å 1 in the neighborhood of the CoO (002) reflection. Even so, the diffuse scattering surrounding these reflections is clearly visible. As shown in Figure 3.18, after the appearance of diffuse intensity in the second region between 130 Å and 216 Å of CoO on MgO, the shape and separation of the side bumps is independent of the thickness d. Therefore, in Figure 3.19 we only present the fit for the 170 Å sample which should be representative for this group. The next region corresponds to the intermediate

38 86 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.18: q x d scans, as shown by the horizontal line in Figure 3.15, for CoO layers on MgO for increasing thickness. The thinnest (115 Å) film shows almost no diffuse scattering. For increasing thickness the central sharp coherent peak disappears, while the diffuse scattering gets more and more pronounced.

39 3.10. Fitting the experimental diffuse scattering data 87 Figure 3.19: Comparison of the experimental with simulated q x d scans calculated using Kaganer s approach in the low density approximation. (a) and (b) Scans trough the (002) and (004) reflection from the 170 Å CoO on MgO can be fitted using the low dislocation density approximation. This region is a meta-stable region where the multiplication of dislocations is hindered by kinetic barriers. (c) At the intermediate density region the side lobes start to disappear. The fit here was done using the intermediate density model. At these thicknesses the film is relaxing and the density of dislocations is rapidly increasing with further increase in thickness. (d) The same scans but for the thickest 860 Å CoO on MgO film. The thick film belongs to the high density region. The fit was done using the high density approximation described in section 3.9 and exemplified in more detail in the following section. From this last fit the linear dislocation density ρ = was deduced. density where ρ d 1 for which the fit for the 356 Å CoO on MgO sample is displayed. This intermediate limit can only be fitted well with ρ d = 2. This means that we overestimate the dislocation density, because this results in ρ 0.012, and this value is almost equal with the misfit f=0.015 for the CoO/MgO system. Possible reasons for this discrepancy are discussed below. The best fit was achieved for the thickest CoO film using the high dislocation

40 88 Chapter 3. X-ray diffraction on thin films and multilayers density limit, for which the value of ρ d = 4.3 was found. Knowing that the thickness of the film d is almost 400 ML, this resulted in a linear dislocation density of ρ= This is a value smaller than the misfit (f =0.015) for the CoO/MgO system which suggests that the film is not entirely relaxed and a small compressive strain remains in the film. This result is in accordance with the out-of-plane c lattice parameter value, measured from the specular scans represented by the last data point in Figure 3.8 (a). The values shown for our best fits are not entirely correct, since one should also take into account the threading dislocations as well as the correlations with other dislocations which this model does not account for. It is very probable that these values are slightly overestimating the real linear dislocation density. Note that in the simulations only the diffuse scattering was calculated. These calculations can be extended also in the perpendicular direction and thus for every q z value a q x scan can be calculated. In this way it is easy to compare the shape of the simulated and measured diffuse scattering. The calculated and measured RSM are presented in Figure 3.20 not only for the specular but also for the (224) off-specular reflection. One can see that although the overall shape is comparable, the number of visible features in the simulation is higher than in the measurement. One of the reasons for this is the insufficient intensity of the X-ray beam of our apparatus. Finally in Figure 3.21 we present another RSM from an off specular reflection and the corresponding calculation, as well as some further simulations of other achievable reflections using the λ = 1.54 Å X-ray wavelength. One should also note that by decreasing the wavelength and thus increasing the diameter of the Ewald sphere, other specular and off-specular reflections can be measured where according to our simulations more structure is expected in the diffuse scattering. An important point, almost always overlooked in reports studying relaxation is concerning the position of the specular Bragg peaks of a partially relaxed film. It is inferred by these authors, without any proof, that the position of the coherent peak follows the mean distortion in the crystal. Kaganer et al. [21] proved though, that this is the case only for the high density limit. At low dislocation densities, although the intensity of the coherent peak decreases exponentially by increasing ρd, the position of the coherent part of the diffracted intensity is not affected by the mean distortion. This is the reason for the discrepancies of the critical thickness resulting from Figures 3.12 and 3.8, and the presence of diffuse scattering because of the dislocations relieving strain already at lower thicknesses. Knowing that the sum of the coherent and diffuse part of the diffracted intensity has to be constant if normalized

41 3.10. Fitting the experimental diffuse scattering data 89 Figure 3.20: Measured and simulated RSMs around specular or off-specular reflections of CoO on MgO. (a) and (b) RSM around the (002) of a 143 A CoO film on MgO and a simulated RSM using the low density limit (eq discussed in section 3.9). (c) and (d) RSM around the (004) reflection of a 216 A CoO film on MgO, and a simulated map using the same equation. Finally a RSM around an off-specular reflection, namely the (224) of the 216 A CoO film on MgO (e) in this case we zoomed in on the film reflection so the substrate MgO (224) is not visible and the corresponding calculated spectra (f).

42 90 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.21: Measured and simulated RSMs around off-specular reflections of CoO on MgO. (a) and (b) RSM around the (204) of a 216 A CoO film on MgO and a simulated RSM using the low density limit eq discussed in section 3.9. (c) and (d) simulated RSM around the (115) and (113) reflections calculated in the low dislocation density limit.

43 3.10. Fitting the experimental diffuse scattering data 91 Figure 3.22: Ratio of integrated intensities from the boxes shown in Figure 3.14 for all thicknesses. the diffuse scattering increases rapidly above 125 Å, and becomes higher than the coherent part after about 130 Å thickness of the CoO layer on MgO. to the thickness of the film, we compare in Figure 3.22 how these intensities develop as a function of thickness. The data points represent integrated intensities from the two boxes depicted in Figure 3.14 normalized to the sum of the coherent and diffuse intensities. Because the intensity scattered in the specular direction (into the narrow box represented in Figure 3.14) is not entirely made of coherently scattered X-rays, this simple integration procedure is not entirely correct. However, the error made is not extremely large, since at low dislocation densities the diffuse intensity in this position of the reciprocal space is vanishingly small. The figure therefore represents just a guideline. If the diffuse scattering was correctly separated, the shape of the curve would be just slightly altered at low thicknesses and the last two points, representing the diffuse and coherent part, would be very close to one and zero, respectively. However, from this analysis one can clearly see that the diffuse scattering becomes a significant fraction of the total intensity for a thickness of 130 Å or higher of the CoO layer on MgO. The sudden change in the intensity distribution above 125

44 92 Chapter 3. X-ray diffraction on thin films and multilayers Å comes as an additional prove confirming that the relaxation process started already between Å thickness of the CoO layer on MgO and that a single specular scan is not sufficient to deduce the state of the relaxation. Data taken with synchrotron radiation would improve very much the statistics, and since, judging from the calculations, off-specular spots have increased number of features around the reflections, the high intensity combined with an increased resolution would be useful for achieving better correspondence between experiment and simulation High density limit In the following we will illustrate the high density limit (ρ d 1). The experimental XRD data was collected from thick layers of MnO epitaxially grown on MgO. The films were covered by a thin CoO layer in order to prevent oxidation. The misfit between bulk MnO and MgO is of the order of 5 %. This large misfit makes this system a good candidate for the high density limit since the relaxation of the strain is rapid, thus ρ is large. Figures 3.23 and 3.24 present two RSM s around the (113) and (224) reflections of the MgO substrate and MnO film, respectively, of a sample having the following composition: 540 ML MnO on MgO covered with 40 ML CoO. The figures show the experimental data, the calculated RSM s as well as the horizontal and vertical line scans presented by the symbols (measured) and by the solid lines (calculated). The calculation of the diffuse scattering was done as explained in Section 3.9. The parameters for the two reflections were the same ρ=0.416, d=60, and ν=0.33. The values of the three quantities w xx, w xz and w zz were evaluated for every z ( [0, d]) by transforming the integrals from equations 3.31 into simple sums which run from ( length 1 2 ) = 100 to ( length 1 2 ) = 100 for x. Increasing the value of length did not change the calculated profiles. The integral over z was also transformed into a sum from 0 to d, and evaluated for every (q x, q z ) pair. From the fits at the bottom of these figures it becomes clear how well the theory describes the experimental measurements for the high density case, although the only parameter which is used is the product ρ d, in our case ρ d = 25. Knowing now the thickness of our sample of 540 ML, which was deduced from the Mn metal flux and also from the fit of the reflectivity curve, one can deduce the real linear dislocation density ρ = 25/540 = 0.046, which corresponds to one dislocation per 21.6 lattice units. This value of ρ is slightly lower than the misfit of the MnO with the substrate, which suggests that the film is not entirely relaxed. This fact is in agreement with the value of the out-of-plane lattice parameter c=4.453 Å, being slightly larger than the bulk value (a MnO = 4.445Å). In order to

45 3.10. Fitting the experimental diffuse scattering data 93 fit the data a trial and error method was used. For this the following general predictions were kept in mind: The shape and orientation of the calculated diffuse intensity will strongly depend on the type of dislocations which are considered in the calculation. Three types of dislocations were considered: Lomer type edge dislocations (b x =1, b y =0, b z =0) which can occur in oxides but only at high temperatures ( C ), normal type edge dislocations (as shown in figure 3.17) (b x =1, b y =0, b z =-1) and 60 dislocations which occur in semiconductors such as Si or Ge (b x = 2 2, b y = 2 2, b z=-1). The peak widths q x,z are proportional to ρ/d and the aspect ratio q x / q z is representative of a particular reflection (Q x, Q y, Q z ) for a particular type of dislocation, or a mixture of dislocations. The aspect ratio q x /q z as well as the orientation of the ellipse shaped diffuse intensity profile is strongly dependent on the value of the Poisson ratio (ν) for a certain type of dislocation or mixture of dislocations. This way one is able to find a particular parameter set in order to simulate, with a certain precision, not only the shape of the diffuse peaks but also their orientation. It is worthwhile to turn our attention towards the orientation of the peaks in the reciprocal space. In the most rudimentary picture, one would imagine the layer as composed of regions having different in-plane stress profiles, whereas the corresponding out-of-plane c lattice parameters are given by the material specific Poisson ratio (ν). In this case one would expect that the scattered diffuse intensity profile generated by such a strained film follows the line connecting the position of the completely strained state with the position of the entirely relaxed state. These points in the reciprocal space are depicted in Figure 3.25 (a). The angle α formed by the horizontal and the line connecting these two points can be calculated once one knows the position of these points as shown in the figure for the case of the (224) reflection: ( ) 4(a MnO α = arctan c MnO ) 2 2(a MgO a MnO ) (3.32) where a MnO = 2π/a MnO, and c is given by the Equation 3.20 (in which one should read MnO instead of CoO and which contains the dependence on the Poisson ratio). This behavior is illustrated in the following two graphs in Figure 3.25 by the gray squares as a function of ν for the (113) and (224)

46 94 Chapter 3. X-ray diffraction on thin films and multilayers Figure 3.23: (a) Reciprocal Space Map around the (113) reflection from a 45 ML CoO on 540 ML MnO bilayer grown on MgO substrate. (b) Calculated RSM using the high density limit method described in Section 3.9. The parameters used for this simulation were ν MnO = 0.33, d=60, ρ = (c) Experimental and simulated scans as presented by the patterns crossing the (113) peak in the top figure, solid lines represent the corresponding simulations. The match is virtually perfect.

47 3.10. Fitting the experimental diffuse scattering data 95 Figure 3.24: (a) Reciprocal Space Map around the (224) reflection from a 45 ML CoO on 540 ML MnO bilayer grown on MgO substrate. (b) Calculated RSM using the high density limit method described in Section 3.9. The parameters used for this simulation were ν MnO = 0.33, d=60, ρ = (c) Experimental and simulated scans as presented by the patterns crossing the (224) peak in the top figure, solid lines represent the corresponding simulations. The match is virtually perfect.